life expectancy of animals in general and mammals and primates?

I thought that in nature there was some kind of general rule that the life expectation of an animal can be determinant by the time it takes to be able to reproduce (sexually mature) times a fixed factor (like say 6).

I was trying to find this information again but I cant seem to find it, maybe i been dead wrong about this? Or maybe it was the time it takes to bear a child?